The point (1,13)

Question

lies on the line f(x)=13x
. The function g(x)=log13x
includes a point of a reflection across the line y=x
that lies on the line g(x)=log13x
. In which quadrant of the coordinate plane is this point located?

Answer 1

To find the point of reflection of \( g(x) = \log_{13} x \) across the line \( y = x \), we first need to determine the relationship between the function and its reflection.

A point \( (a, b) \) on the curve \( y = g(x) \) has its reflection across the line \( y = x \) at the point \( (b, a) \). Thus, we want to find \( (x, g(x)) = (x, \log_{13} x) \) and then find the reflected point \( (g(x), x) = (\log_{13} x, x) \).

We need to examine two characteristics:

1. The output \( g(x) = \log_{13} x \).
2. The input \( x \) and the condition that it must satisfy, which is that the reflected point must also lie on the line \( g(x) = \log_{13} x \).

Thus we set up the equation for the reflection:

\[ g(g(x)) = g(\log_{13} x). \]

We need to calculate \( g(g(x)) \). Using the property of logarithms:

\[ g(g(x)) = g(\log_{13} x) = \log_{13}(\log_{13} x). \]

Now we need to ensure that:

\[ \log_{13} x = x. \]

However, this equality doesn't lead us easily to recognizable values. Instead, we need to assess the values in different quadrants.

The logarithmic function \( g(x) = \log_{13} x \):

• Is defined for \( x > 0 \) and yields values across all quadrants.
• Will return positive values for \( x > 1 \) and negative for \( 0 < x < 1 \).

To find out where reflections fall, let's examine \( (x, g(x)) \) for progressive \( x \):

• For \( x = 1 \): \( g(1) = 0 \) → Point is \( (1, 0) \) → Reflection is \( (0, 1) \) (Quadrant II)
• For \( x = 13 \): \( g(13) = 1 \) → Point is \( (13, 1) \) → Reflection is \( (1, 13) \) (Quadrant I)
• For values \( 0 < x < 1 \): \( g(x) < 0 \) leads to negative reflections in Quadrant II.
• The point \( (b, a) = (\log_{13} x, x) \) may yield both negative and positive outputs, indicating quadrant-specific results.

Next, we check when their resulting outputs would ensure they preserve quadrant behavior:

• If \( 0 < x < 1 \), \( (g(x), x) \) places in Quadrant II (negative y).
• If \( x = 13\), \( (1, 13) \) remains entirely within Quadrant I.

The logarithmic function reveals that reflected points \( (\log_{13} x, x) \) primarily yield outputs that appear across Quadrant I for \( x > 1\) while revealing behavior across Quadrant II as \( x\) approaches 0.

In conclusion, depending on resulting \( x\) values and logarithmic properties, the reflection point \( (g(x), x) \) correctly lands within:

Quadrant I for real positive intervals of reflected outputs.